Assimilation iis the process whereby persons and groups acquire the culture of other group in which they come to live, by adopting its attitudes and values its pattern of thinking and behaving.
Cultural assimilation is when an individual or individuals adopts some or all aspects of a dominant culture (such as its religion, language, norms, values etc.). Cultural assimilation is a process of socialization. It can be a voluntary process, but can also sometimes be the result of involuntary political decisions.
A group (a state or an ethnicity) can spontaneously adopt a different culture due to its political relevance, or to its perceived superiority. The first is the case of the Latin culture and language, that were gradually adopted by most of the subjugated people.
The second is the case of subjugated, but older and richer culture, which see itself imitated by the new masters, e.g. the victorious Roman Republic adopted more from the Hellenistic cultures than it imposed in most domains, except such Roman specialities as law and the military.